How do you evaluate the definite integral by the limit definition given #int 4dx# from [0,3]?

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Answer 1 · 2017-05-19T08:03:09

#12#

#int_0^3 4 dx = [4x]_0^3#

# = [4*3] - [4 * 0] = 12#

Answer 2 · 2017-05-22T03:18:32

see below

#int_0^3 4 dx#,

Integral by limit definition is given by

#lim_(n ->oo) sum_i^n f(c_i) delta x#

therefore,

#delta x = (3-0)/n = 3/n#
#c_i = a + delta x = 0 + 3/ni = 3/ni#

#int_0^3 4 dx= lim_(n ->oo) sum_i^n f(c_i) delta x#,

#= lim_(n ->oo) sum_i^n f(3/ni) 3/n #

for #i = 1,2,3,...,n -> f(x) =4#

#= lim_(n ->oo) sum_i^n 4 *3/n = lim_(n ->oo) sum_i^n 12/n#

# =lim_(n ->oo) n 12/n = lim_(n ->oo) 12 = 12 #